Unit 7B

Integrals Part 2

Lesson 1: Distance, Displacement, and Velocity

Instantaneous and Average Rates of Change

In Unit 2 – Derivatives, the concept of limits was explored to find the slope of a function at any point. This Lesson explores the relationship between slope and instantaneous rate of change.

Note: In this video, the narrator uses the phrase goes to. Recall the proper terminology for this is approaches.

Watch the video Instantaneous and Average Rates of Change to review this relationship.

In the video, first principles was used to find the instantaneous rate of change. Another way to find the instantaneous rate of change is to find the derivative of the given function, and then evaluate the derivative at the specified point.

Example (from the video)

For the function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mi»x«/mi»«/math», what is the instantaneous rate of change at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/math»?

Solution

First, find the derivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»+«/mo»«mi»x«/mi»«/math».

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/math»

Now, find the instantaneous rate of change at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/math».

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«msup»«mfenced»«mn»1«/mn»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«/mtd»«/mtr»«/mtable»«/math»

This is the same instantaneous rate of change calculated in the video using first principles.

The formulas for average rate of change and instantaneous rate of change were also given in the video.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»m«/mi»«mrow»«mi»a«/mi»«mi»v«/mi»«mi»e«/mi»«/mrow»«/msub»«mo»=«/mo»«mfrac»«mrow»«mo»§#916;«/mo»«mi»y«/mi»«/mrow»«mrow»«mo»§#916;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»§#8722;«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mrow»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»§#8722;«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfrac»«/math»

and

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»m«/mi»«mrow»«mi»i«/mi»«mi»n«/mi»«mi»s«/mi»«mi»t«/mi»«/mrow»«/msub»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«msub»«mi»m«/mi»«mrow»«mi»a«/mi»«mi»v«/mi»«mi»e«/mi»«/mrow»«/msub»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mi»f«/mi»«mfenced»«mrow»«mi»a«/mi»«mo»+«/mo»«mi»h«/mi»«/mrow»«/mfenced»«mo»§#8722;«/mo»«mi»f«/mi»«mfenced»«mi»a«/mi»«/mfenced»«/mrow»«mi»h«/mi»«/mfrac»«/math» or «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»m«/mi»«mrow»«mi»i«/mi»«mi»n«/mi»«mi»s«/mi»«mi»t«/mi»«/mrow»«/msub»«mo»=«/mo»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mo»(«/mo»«mi mathvariant=¨normal¨»a«/mi»«mo»)«/mo»«/math»